Intelligence within Bounds: Why Cognition Requires a Closed Convex Hull

How do discrete symbolic representations emerge from continuous neural dynamics?Unlike Bayesian approaches that presuppose prior distributions, we propose Clock-Selected Compression Theory (CSCT), an axiomatic framework grounded in five principles: (1) cognition operates on continuous streams, (2) representations are constrained to convex combinations within a simplex, (3) discrete events emerge via phase-locked clock selection, (4) irreversible anchors impose thermodynamic directionality, and (5) syntax emerges as geometric interpolation within the semantic convex hull rather than symbol concatenation.Through nine experiments using a minimal neural architecture (the CSCT Engine), we demonstrate that: discretization emerges reliably from continuous dynamics; irreversible anchors outperform self-referential systems in long-term stability; feature binding arises from shared phase without explicit concatenation; semantic grounding requires convex-hull membership (96.7\% vs.\ 16.7\% success for in-hull vs.\ out-hull); and syntactic composition emerges as barycentric interpolation rather than algebraic abstraction.We identify a phenomenon we term \emph{Ungrounded Symbol Acquisition}---discrete codes assigned without reconstructable meaning---providing a mathematical instantiation of the symbol grounding problem.These findings suggest that bounded geometric constraints, rather than unbounded scaling, may be necessary for stable, meaningful cognition.We discuss implications for understanding neural manifolds, the limitations of self-attention architectures, and testable predictions for neurophysiological validation.